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L"#k=l#k". Proof. J, the above relations hold trivially on (- oo, OJ. }® k(t:} dt:. 0 (a) tE (0, oo). Then 27 Space-time. Sect. 11. Thus, letting -r =A. ) dA. = l#k"'(t). :~+ fl(t--r) (8) kcx(-r) d-r. 0 Because of (a), (c) yields the second relation in (2). (c) D 11. Space-time. We call the four-dimensional point space @"(4) =@"X ( - oo, oo) space-time; in our physical context a point ; = (:ll, t) E @"( 4) consists of a point :1l of space and a time t. The translation space associated with @"(4) is 1"'(4) ='Y'X ( - oo, oo); elements of 1"'(4) will be referred to as four-vectors.

It follows from the spectral decomposition for E that 3* M. E. GuRTIN: The Linear Theory of Elasticity. Sect. 13. where Thus if we let where Po (;;c) =;;c -;;co, the decomposition (i) follows. On the other hand, if we let ua =i(tr E) p 0 , Uc = [E -l(tr E) 1] p 0 , then ua is a uniform dilatation, uc is an isochoric pure strain, and ua+ue=EPo=U. D (6) Decomposition theorem for simple shears. Let u be a simple shear of amount" with respect to the direction pair (m, n). Then u admits the decomposition u=u++u-, where u"' is a simple extension of amount ±" in the direction 1 V2 (m±n).

Let Ua be a class C3 field on R, and let (i) Then (ii) Conversely, let EafJ (=Epa) be a class CN (N ~ 2) field on R that satisfies (ii). Then there exists a class CN+l field Ua on R such that (i) holds. 1 This explicit solution is due to CESARO [1906, 2]. In this connection, see also VoLTERRA [1907, 4], SOKOLNIKOFF [1956, 12], BOLEY and WEINER (1960, 3]. 42 M. E. GuRTIN: The Linear Theory of Elasticity. Sect. 15. Proof. That (i) implies (ii) follows upon direct substitution. To prove the converse assertion assume that (ii) holds.

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